Is there a rule or a scale for judging if a certain MSE is small, very small, big or very big? Please give also a source to underpin your recommendation.
There is no such rule and there will never be one. MSE, like many other similar criteria is a relative measure (like many other things in statistics, say correlation, or size of confidence intervals). It uses the same units (squared) as your data.
To simplify the example, let's say that we are talking about mean absolute error. Imagine that your variable is distance in meters, you have some model that outputs predictions that on average differ by thousands of meters -- is MAE big or small? Now imagine that you recoded your data into kilometers, now MAE is just few kilometers -- is is "smaller"? It is the same with MSE, but the units are squared.
You can check also the thread that discusses similar case of interpreting the size of standard deviation.